求关于X的函数Y=(a+sinx)(a+cosx)(a>0)的最大值与最小值

f(x)=(a+sinx)(a+cosx)
=a^2+a(sinx+cosx)+sinxcosx
=a^2+√2asin(x+π/4)+1/2sin2x
=a^2+√2asin(x+π/4)-1/2cos(2x+π/4)
=a^2+√2asinθ-1/2cos2θ ,(θ=x+π/4)
=a^2+√2asinθ-1/2[1-2sin^2θ]
=t^2+√2at+a^2-1/2 ,(t=sinθ)
=(t+√2/2*a)^2+(a^2-1)/2 ，(-1≤t≤1)
当a≥√2时
f(t)≤f(1)=a^2+√2a+1/2
f(t)≥f(-1)=a^2-√2a+1/2
当0≤a≤√2
f(t)≥f(-√2/2*a)=(a^2-1)/2
f(t)≤f(1)=a^2+√2a+1/2

∴f(x)的最大值为a^2+√2a+1/2
当a≥√2时，
f(x)的最小值为a^2-√2a+1/2
当0≤a≤√2时
f(x)的最小值为（a^2-1)/2